A further interpretation of Schwarz preconditioning in relation to immersed finite element methods can be obtained from the additive Schwarz lemma. This lemma states that for a Symmetric Positive Definite (SPD) matrix \(\mathbf {A}\) , the \(\mathbf {B}_{\textsc {AS}}^{-1}\) -inner product of an arbitrary vector \(\mathbf {y}\)  is equal to [1]}, [2]} (see [3]}, [4]} for this specific form)
